Edward DeMarco, acting director of the Federal Housing Finance Agency, during an interview last year.

Fannie Mae and Freddie Mac said Thursday that they’d be sending $39 billion to the U.S. Treasury this quarter, which means by the end of the year, they’ll be very close to having sent nearly as much in dividend payments to the government as the amounts the government was forced to invest in them—nearly $188 billion—during the financial crisis.

But the regulator that oversees the companies isn’t terribly enthralled by some of the side-effects of this profitability. Namely, he has raised concerns that profits are allowing some interest groups to have amnesia about the failure of the mortgage-finance giants and their business model five years ago.

“I think some people are forgetting a little history,” said Edward DeMarco, the acting director of the Federal Housing Finance Agency, in a meeting with reporters two weeks ago.

At an industry conference last month, Mr. DeMarco said that “memories may begin to fade” as the crisis becomes more distant. While the companies are now producing profits, that shouldn’t change the fact that the firms’ business model remains broken, he said.

“As Fannie Mae and Freddie Mac have begun to report positive net income, there may be a growing perception that the problems that led to conservatorship have been fixed,” he said in the speech. “That is not the case.”

Chief among those unaddressed problems: Fannie and Freddie long benefited from a fuzzy “implied” guarantee, which is a major reason the government was forced to rescue the firms during the crisis. Those guarantees were made nearly explicit after the crisis, and any overhaul will have to resolve what to do with the existing debt guarantees and any future system of guaranteeing mortgage-backed securities.

In his speech, Mr. DeMarco said it wasn’t clear how much of the benefit of those guarantees went to borrowers as opposed to the companies’ shareholders and executives. Moreover, he said, to the extent that those guarantees reduced borrowing costs for home buyers, they “surely resulted in higher house prices,” benefiting existing owners over prospective ones.

Mr. DeMarco said that until Congress decides on the future of the housing-finance market, he will continue to press ahead with reorganizing and “winding up the affairs” of the companies. He outlined four key steps:

First, Fannie and Freddie will continue to increase the fees that they charge lenders for backing and securitizing mortgages. The FHFA has directed both companies to increase fees several times, on top of an increase that was mandated in 2011 by Congress.

Second, the firms will move ahead with plans to sell off some of the credit risk on mortgages that they guarantee, using new derivatives.

Third, the FHFA will press on with previously announced plans to gradually reduce the maximum loan limits that the companies can purchase, a move that has stoked significant opposition from lawmakers that represent more expensive housing markets, not to mention the real-estate industry. The agency has also mandated the firms to shrink by 10% the amount of multifamily mortgages that they purchased, compared to last year.

Finally, the agency will reposition parts of Fannie and Freddie by setting up a joint-venture controlled by the companies and the regulator to oversee the creation of a new platform for securitizing mortgages.